- Email: [email protected]
Reply to: A note on some neuroimaging study of natural language quantifier comprehension
Neuropsychologia 45 (2007) 2161
Letter to the Editor Reply to: A note on some neuroimaging study of natural language quantifier comprehension We agree with Szymanik that the correlation between quantifier class and working memory is, at best, imperfect. Indeed, it is straightforward to construct an argument that the class of natural language quantifiers extends well beyond the class of languages accepted by push-down automata. Consider the following sentence: An equal number of doctors, lawyers and dentists play golf. This example could only be verified by an automaton in the class of embedded push-down automata (EPDA), a class of automata that are more powerful still than PDA (Clark & Morton, 1999). Thus, it is clear that the association between quantifier class and automata is imperfect, although suggestive. A better hypothesis is that quantifier class correlates with demands on executive resources. Consider Szymanik’s example of “most” as in: Most As are B. The algorithm he describes is similar to the Boyer-Moore vote counting algorithm which pairs off opponents to determine majorities (Boyer & Moore, 1991). This algorithm has relatively low complexity but does demand some executive resources to organize the array and cancel out pairs. As usual, mathematics provides a number of different methods for analyzing various problems within the large and varied class of quantifiers. Studies of brain-behavior relationships cannot achieve the fine-grained characterizations necessary to test these detailed distinctions in a rigorous manner. For example, the area activated in the imaging study spans inferior frontal cortex, an area important for working memory, and dorsolateral prefrontal cortex, a region often associated with strategic planning and organization. This would also be consistent with our observations of patients with focal neurodegenerative diseases such as corticobasal degeneration (CBD) and frontotemporal dementia (FTD) (McMillan, Clark, Moore, & Grossman, 2006). Patients with CBD have significant disease of the parietal lobe and the frontal lobe, while patients with FTD have frontal lobe disease but rarely have disease compromising the parietal lobe. We find
0028-3932/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.neuropsychologia.2007.03.001
that CBD patients have difficulty understanding first-order quantifiers that depend on number knowledge, although FTD patients have no difficulty with first-order quantifiers. This corresponds to the fMRI study associating first-order quantifiers with parietal cortex. Patients with FTD and CBD both have difficulty with higher-order quantifiers, presumably related to their frontal disease. These patients also have difficulty on measures of executive resources, and difficulty on measures of executive resources correlates only with their comprehension of higher-order quantifiers. None of this is specific enough to address the mathematical distinction between PDA and EPDA, assuming that this is the correct characterization of the distinction. More central to our work is the principle that mathematical objects like complexity classes can help constrain interpretations of the organization and functioning of the brain, and that the brain’s apparent solutions for these various cognitive challenges can help constrain the variety of mathematical objects that should be entertained in this context. References Boyer, R. S., & Moore, J. S. (1991). MJRTY—a fast majority vote algorithm. In R. S. Boyer (Ed.), Automated reasoning: Essays in honor of Woody Bledsoe (pp. 105–117). Dordrecht, the Netherlands: Kluwer Academic Publishers. Clark, R., & Morton, T. (1999). A note on a certain class of quantifier denotations in natural language. Mathematics of Language, 6. McMillan, C., Clark, R., Moore, P., & Grossman, M. (2006). Quantifier comprehension in corticobasal degeneration. Brain and Cognition, 62, 250–260.
Robin Clark Corey McMillan Murray Grossman ∗ University of Pennsylvania Medical Center, Department of Neurology, 3400 Spruce Street 3 West Gates, Philadelphia 19104-4283, United States ∗ Corresponding
author. Tel.: +1 215 662 3361; fax: +1 215 349 8464.
E-mail address: [email protected] (M. Grossman) Available online 4 March 2007
Letter to the Editor Reply to: A note on some neuroimaging study of natural language quantifier comprehension We agree with Szymanik that the correlation between quantifier class and working memory is, at best, imperfect. Indeed, it is straightforward to construct an argument that the class of natural language quantifiers extends well beyond the class of languages accepted by push-down automata. Consider the following sentence: An equal number of doctors, lawyers and dentists play golf. This example could only be verified by an automaton in the class of embedded push-down automata (EPDA), a class of automata that are more powerful still than PDA (Clark & Morton, 1999). Thus, it is clear that the association between quantifier class and automata is imperfect, although suggestive. A better hypothesis is that quantifier class correlates with demands on executive resources. Consider Szymanik’s example of “most” as in: Most As are B. The algorithm he describes is similar to the Boyer-Moore vote counting algorithm which pairs off opponents to determine majorities (Boyer & Moore, 1991). This algorithm has relatively low complexity but does demand some executive resources to organize the array and cancel out pairs. As usual, mathematics provides a number of different methods for analyzing various problems within the large and varied class of quantifiers. Studies of brain-behavior relationships cannot achieve the fine-grained characterizations necessary to test these detailed distinctions in a rigorous manner. For example, the area activated in the imaging study spans inferior frontal cortex, an area important for working memory, and dorsolateral prefrontal cortex, a region often associated with strategic planning and organization. This would also be consistent with our observations of patients with focal neurodegenerative diseases such as corticobasal degeneration (CBD) and frontotemporal dementia (FTD) (McMillan, Clark, Moore, & Grossman, 2006). Patients with CBD have significant disease of the parietal lobe and the frontal lobe, while patients with FTD have frontal lobe disease but rarely have disease compromising the parietal lobe. We find
0028-3932/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.neuropsychologia.2007.03.001
that CBD patients have difficulty understanding first-order quantifiers that depend on number knowledge, although FTD patients have no difficulty with first-order quantifiers. This corresponds to the fMRI study associating first-order quantifiers with parietal cortex. Patients with FTD and CBD both have difficulty with higher-order quantifiers, presumably related to their frontal disease. These patients also have difficulty on measures of executive resources, and difficulty on measures of executive resources correlates only with their comprehension of higher-order quantifiers. None of this is specific enough to address the mathematical distinction between PDA and EPDA, assuming that this is the correct characterization of the distinction. More central to our work is the principle that mathematical objects like complexity classes can help constrain interpretations of the organization and functioning of the brain, and that the brain’s apparent solutions for these various cognitive challenges can help constrain the variety of mathematical objects that should be entertained in this context. References Boyer, R. S., & Moore, J. S. (1991). MJRTY—a fast majority vote algorithm. In R. S. Boyer (Ed.), Automated reasoning: Essays in honor of Woody Bledsoe (pp. 105–117). Dordrecht, the Netherlands: Kluwer Academic Publishers. Clark, R., & Morton, T. (1999). A note on a certain class of quantifier denotations in natural language. Mathematics of Language, 6. McMillan, C., Clark, R., Moore, P., & Grossman, M. (2006). Quantifier comprehension in corticobasal degeneration. Brain and Cognition, 62, 250–260.
Robin Clark Corey McMillan Murray Grossman ∗ University of Pennsylvania Medical Center, Department of Neurology, 3400 Spruce Street 3 West Gates, Philadelphia 19104-4283, United States ∗ Corresponding
author. Tel.: +1 215 662 3361; fax: +1 215 349 8464.
E-mail address: [email protected] (M. Grossman) Available online 4 March 2007